This question is from "The Mathematics of Poker". I have three equations based off a 3x3 Rock, Paper, Scissors payout table where winning with scissors is given a +2 payout instead of +1.

Equations for Payoffs of choosing rock, paper, and scissors for player one, in order:
-b+c
a-2c
-a+2b

Solution:
a=2b=2c

Can someone show me how the solution was reached?

Ironically, I am interested in this for reasons outside of poker. It is just cool what all you can turn into math.

Thanks for the help

You need to find a Nash equilibrium. You can use a zero-sum game solver:
http://levine.sscnet.ucla.edu/games/zerosum.htm

I've no experience with this but:

-b+c=a-2c (1)
a-2c=-a+2b (2)

(1) a=-b+3c
(2) a=b+c

So b+c=-b+3c => b=c
So a = 2b = 2c
Frequency: a is 0.5 and b is 0.25 and c is 0.25

I guess something like this.

I guess my approach is incorrect ... it's probably ok to do it this way in this case (just because the solution is ok ) but not suitable to solve other pay-out tables.

When you did the first steps of:

-b+c=a-2c (1)
a-2c=-a+2b (2)

did you set those equations equal to each other because that it is a given that this is the first step in how you are supposed to solve these systems of equations?


Also in:
b+c=-b+3c => b=c

How did you get b=c? Substitution, I'm guessing.

and how did you use b=c to find:
a = 2b = 2c

I am not sure why b and c have coefficients of 2.

this should say
b + c = -b + 3c

add b to each side, subtract c
2b = 2c

Out of some dumb-assing around I discovered this.

As for other games, I know the answer is a=b=c for the standard weighted rock, paper, scissors game going by "The Mathematics of Poker", but I cannot get that.

I come up with this when I try to work it:
-b+c
a-c
-a+b

1) -b+c=a-c => a=-b+2c
2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c

-b+2c=1/2b+1/2c
-2b+4c=b+c
-3b=-3c => b=c
3b=3c => b=c
so,
a=3b=3c

the correct answer is a=b=c according to the book. The equation in the up-top example of 2b=2c was not simplified to b=c, so I am pretty sure not simplifying at the end is not the problem.

Where did I mess up?

Sorry, the Physics Forum is just not responding.

Thanks again

According to this a=b=c

a=-b+2b=b or a=-c+2c=c
a=1/2b+1/2b=b or a=1/2c+1/2c=c

But keep in mind I'm pretty sure this method is not suitable to solve all pay-out tables. I don't know where I use a shortcut. Better use the solver in the link I provided instead.

Substituting b=c into either 1) or 2) shows a=b or a=c as needed

I think I missed something when a pay-out table is asymmetric.

I guess for symmetrical pay-out tables (like OP gave up as examples) it's easy:
a, b and c are the same for both players.
=> You can use my method:
Just make the equations (1) and (2) and solve it.
It's easy to understand, expressions need to be equal to eachother and zero because of the symmetry.
Example: You can't win/lose playing %Rock while your opponent loses/wins with %Rock. And if you both are winning or losing you are not in a Nash equilibrium (you are both vulnerable to exploitation).

If you have asymmetrical pay-out tables it's different:
a, b and c are different for both players.
=> You can't simply use my method:
Equations in (1) and (2) cannot be made.

When I gave an example of the first rock paper scissor game where the scissors had a weight of two the answer was

a=2b=2c
Why were b and c not simplified to b=c in the final answer, so that it was a=b=c? If 2b=2c, b=c.

But,
In the ordinary rock paper scissors game I worked where the answer was a=b=c :

I got a=3b=3c and OmahaDonk, said that my answer was correct, because I could plug b=c into either equation
1) -b+c=a-c => a=-b+2c
2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c
which means that 3b=3c is simplified.

What am I missing? I would have worked them both like Omaha, but then would have missed the first answer because I simplified.

Man, I am missing something little here.

Thanks for the patience from the two of you.

Just make the correct substitutions (reread my 7th post).

If a=-b+2c and b=c => a=-b+2b=b or a=-c+2c=c => a=b=c
If a=b+c and b=c => a=b+b=2b or a=c+c=2c => a=2b=2c